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In this paper, a high-precision eigenmode-computation analysis of arbitrary azimuthal mode numbers in periodic disc-loaded structure based on variational method will be discussed. It allows for rounding the edge of a disk hole without any approximation in shape treatment and calculates the modes exactly synchronous to the beam. It converges much faster than the mesh-based computer code. Good agreement was observed between the results of variational method and those of other methods.

The performance of existing and proposed collider is often limited by the beam-beam interaction. The tune shift and spread from head-on and long-range beam-beam interactions for both round and flat Gaussain beam have been studied with analysis method in this paper. These formulas are useful for multi-bunch colliders where both short-range and long-range beam-beam interactions should be considered.

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